1. Field of the Invention
The invention is generally directed to technology to track, and optionally to control, an observed system or “target” or a platform for observing the target. Embodiments of the invention can be applied to track virtually any kind of target, including air, space, water or land-borne vehicles, missiles, munitions, astronomical phenomena such as stars, planets, moons, asteroids, meteors, or biological phenomenon such as microorganisms or cells, etc. The invention can also be applied to a vehicle or machine for use in obstacle avoidance. Embodiments of the invention can be applied to control the target such as in the context of the target being a robotic arm or machine part. Other embodiments can be applied to control a platform such as a vehicle or the like, to keep a target under observation.
2. Description of the Related Art
The problem of state estimation of nonlinear stochastic systems is a widely encountered problem in science and engineering and has received a considerable amount of attention since the early development of methods for linear state estimation. [1] New directions in nonlinear observer design. In H. Nijmeijer and T. Fossen, editors, Lecture Notes in Control and Information Sciences 244. Springer Verlag, London, 1999; [2] A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Syst. Contr. Letters, 3:47, 1983; [3] D. Bestle and M. Zeitz. Canonical form observer design for nonlinear time variable systems. Int. J. Control, 38:419-431, 1983; [4] S. Nicosia, P. Tomei, and A. Tomambe. An approximate observer for a class of nonlinear systems. Syst. Contr. Letters, 12:43, 1989; [5] H. K. Khalil and A. N. Atassi, Separation results for the stabilization of nonlinear systems using different high gain observer designs. Syst. Contr. Letters, 39:183-191, 2000. In the case of linear dynamical systems with white process and measurement noise, the filter is known to be an optimal estimator. [6] Emil Kalman, Rudolph. A new approach to linear filtering and prediction problems. Transactions of the ASME—Journal of Basic Engineering, 82(Series D):35-45, 1960, while for linear systems with deterministic disturbances the Luenberger observer offers a complete and comprehensive solution for the problem of state estimation. [7] D. G Luenberger. Observing the state of a linear system. IEEE Trans. Milit. Electr., 8:74, 1963. Of the numerous attempts being made for the development of nonlinear estimators, the most popular one is the extended Kalman filter (EKF), whose design is based on a first order local linearization of the system around a reference trajectory at each time step. [8] A. H. Jazwinski. Stochastic Processes and Filtering Theory. New York: Academic, 1970; [9] Robert Grover Brown and Patrick Y. C. Hwang. Introduction to Random Signals and Applied Filtering. John Wiley and Sons, Inc., 1992; [10] P. Zarchan. Tactical and Strategic Missile Guidance (Second Edition), volume 57. AIAA, 1994. In addition to its application in the field of nonlinear state estimation, EKFs are also used to estimate the unknown parameters of stochastic linear dynamical systems as reported in [11] H. Cox. On the estimation of state variables and parameters for noisy dynamical systems. IEEE Transactions on Automatic Control, AC-9:5-12, February 1964; [12] K. J. Astrom and P. Eykhoff. System identification—a survey. Automatica, 7:123-162, 1971; [13] A. P. Sage and C. D. Wakefield. Maximum likelihood identification of time varying and random system parameters. Int. J. Contr., 16(1):81-100, 1972; [14] Lennart Ljung. Asymptotic behaviour of the extended Kalman filter as a parameter estimator for linear systems. IEEE Transactions on Automatic Control, AC-24(1):36-50, February 1979; [15] V. Panuska. A new form of the extended Filter for parameter estimation in linear systems with correlated noise. IEEE Transactions on Automatic Control, AC-25(2):229-235, April 1980. Although parameter estimation of linear and nonlinear systems using EKFs has received a fair amount of attention, nonlinear state estimation using EKFs has become one of the most researched problems. [16] L. Ljung. Analysis of recursive, stochastic algorithms. IEEE Transactions on Automatic Control, AC-22:551-575, August 1977; [17] H. Weiss and J. B. Moore. Improved extended Filter design for passive tracking. IEEE Transactions on Automatic Control, AC-25(4):807-811, August 1980; [18] M. Boutayeb, H. Rafaralahy, and M. Darouach. Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems. IEEE Transactions on Automatic Control, 42(4):581-586, April 1997; [19] Konrad Reif, Stefan Gunther, Engin Yaz, and Rolf Unbehauen. Stochastic stability of the discrete-time extended Kalman filter. IEEE Transactions on Automatic Control, 44(4):714-728, April 1999; [20] Sergio Bittani and Sergio M. Savaresi. On the parameterization and design of an extended Kalman filter frequency tracker. IEEE Transactions on Automatic Control, 45(9):1718-1724, September 2000.
One important application of the EKF to date lies in the realm of target tracking-trajectory estimation and in the area of missile target interception. In [21] R. P. Wishner, R. E. Larson, and M. Athans. Status of radar tracking algorithms. Proc. Symp. on Nonlinear Estimation Theory and Its Appl., San Diego, Calif., :32-54, September 1970, the authors have developed a set of tracking algorithms that are applicable for ballistic reentry vehicles, tactical missiles and airplanes. In [22] Robert A. Singer. Estimating optimal tracking filter performance for manned maneuvering targets. IEEE Transactions on Aerospace and Electronic Systems, AES-6(4):473-483, July 1970; [23] Per-Olof Gutman and Mordekhai Velger. Tracking targets using adaptive filtering. IEEE Transactions on Aerospace and Electronic Systems, 26(5):691-699, September 1990; [24] Taek Lyul Song. Observability of target tracking with range-only measurements. IEEE Journal of Ocean Engineering, 24(3):383-387, July 1999, the feasibility of target tracking is studied from a point of view of range-only measurements. However in some situations it may be impractical to measure the range, and state estimation using measurements of the line-of-sight or bearing angle is highly desirable. Hence, designing EKFs for target trackers with bearings-only measurement has been a widely studied subject. [25] C. B. Chang. Ballistic trajectory estimation with angle-only measurements. IEEE Transactions on Automatic Control, AC-25(3):474-480, June 1980; [26] S. C Nardon and V. J. Aidala. Observability criteria for bearings-only target motion analysis. IEEE Transactions on Aerospace and Electronic Systems, AES-17(2):162-166, March 1981; [27] Vincent J. Aidala and Sherry E. Hammel. Utilization of modified polar coordinates for bearings-only tracking. IEEE Transactions on Automatic Control, AC-28(3):283-294, March 1983; [28] Chaw-Bing Chang and John A. Tabaczynski. Application of state estimation to target tracking. IEEE Transactions on Automatic Control, AC-29(2):98-109, February 1984; [29] T. L. Song and J. L. Speyer. A stochastic analysis of a modified gain extended Kalman filter with applications to estimation with bearings-only measurement. IEEE Transactions on Automatic Control, AC-30(10):940-949, October 1985; [30] Taek Lyul Song. Observability of target tracking with bearings-only measurements. IEEE Transactions on Aerospace and Electronic Systems, 32(4):1468-1472, October 1996. In the case of bearing measurements the process may be unobservable unless the sensing vehicle executes a maneuver [26] S. C Nardon and V. J. Aidala. Observability criteria for bearings-only target motion analysis. IEEE Transactions on Aerospace and Electronic Systems, AES-17(2):162-166, March 1981, which further complicates the bearings-only problem.
A key to successful target tracking lies in the effective extraction of useful information about the target's state from observations. In the setting of estimation, this necessitates adding additional states to model the target dynamics. Consequently, the accuracy of the estimator depends on the accuracy to which the target behavior has been characterized. Target behavior not captured by modeling introduces estimation bias, and can even cause divergence in the estimate. To account for modeling errors in the process, neural network (NN) based adaptive identification and estimation schemes have been proposed. [31] Y. Kim, F. L. Lewis, and C. Abdallah. A dynamic recurrent neural network based adaptive observer for a class of nonlinear systems. Automatica, 33(8):1539-1543, 1998; [32] J. S. J. Lee and D. D. Nguyen. A learning network for adaptive target tracking. IEEE International Conference on Systems, Man and Cybernetics, pages 173-177, November 1990; [33] William A. Fisher and Herbert E. Rauch. Augmentation of an extended Filter with a neural network. IEEE, pages 1191-1196, 1994; [34] Stephen C. Stubberud, Robert N. Lobbia, and Mark Owen. An adaptive extended Kalman filter using artificial neural networks. Proceedings of the 34th Conference on Decision and Control, pages 1852-1856, New Orleans, La.—December 1995; [35] N. Hovakimyan, A. J. Calise, and V. Madyastha. An adaptive observer design methodology for bounded nonlinear processes. Proceedings of the 41st Conference on Decision and Control, 4:4700-4705, December 2002. In [31], an approach is developed that augments a linear time invariant filter with an NN while in [32]-[35], schemes for augmenting an EKF with an NN are provided. However the approaches in [31]-[34] require knowledge of the full dimension of the system. In [35], an approach that does not require the knowledge of the full dimension of the system is developed. However, this approach only permits the augmentation of a steady state Kalman filter with an NN. Since it would be desirable to address target tracking problems which do not have a good model for the behavior of the target, it would be desirable to provide augmentation of an EKF with an NN which accounts for the unmodeled dynamics of the target and platform used to observe the target. Other background information is provided in [39] A. Isidori. Nonlinear Control Systems, Springer, Berlin, 1995.